Difference between "Not Defined" and "Infinity" [Division by zero]
Okay, let's see if we have confusion about this concept: 
What is the result that you get when you divide a number by zero? Is it "infinity" or is it "not defined"?
Is there any difference between the two?
Or Infinity is not defined? ๐
Update:
Let me try to offer my thoughts on this topic.
In the realm of mathematics and engineering, the notion of dividing by zero is a welltrodden path that leads to undefined territory.
When we delve into the act of division, we're essentially inquiring about the number of times one quantity can be contained within another.
However, when the divisor plunges to zero, the question becomes inherently flawed, thus yielding an undefined result.
Infinity, on the other hand, is a concept that embodies an unbounded quantity. While it's not a real number that we can perform conventional arithmetic with, it's a useful idea in many fields, expressing the notion of a quantity without bound.
Now, when you attempt to divide a nonzero number by zero, some may be tempted to proclaim the result as infinity due to the boundless nature of the division.
However, this is a misstep, as infinity is more of a concept rather than a concrete number. The operation is actually undefined in standard arithmetic.
The confusion often stems from the limit scenario, where we examine the behavior of a function as it approaches a certain point.
For instance, as we divide 1 by a number that approaches zero, the result shoots towards infinity. Yet, this is a journey towards infinity, not a concrete arrival at it.
Thus, in summary, the act of dividing by zero is undefined within the standard arithmetic framework.
Infinity and "undefined" are distinct terms with different implications, and the relationship between them in this context is a reflection of the limit scenario, not a direct equivalence.
So, when faced with division by zero, it's prudent to stick with "undefined" as the rightful descriptor.
Replies

Anil Jain
There are two concepts...The_Big_KOkay, let's see if we have confusion about this concept: 
What is the result that you get when you divide a number by zero? Is it "infinity" or is it "not defined"?
Is there any difference between the two?
Or Infinity is not defined? ๐
Let me try to explain this...
Something divided by 0 would always be 'undefined', but my question is is 'Infinity' is defined. We assume that countless figure is 'Infinity', but isn't that undefined again.
Anways in Layman's language why something divided by 0 is undefined. I found beautiful excerpt as follows:
twelve divided by three:
You have twelve cookies to share equally among three children, how many cookies does each child get?
five divided by one:
You have five cookies to share equally among one child, how many cookies does that child get?
zero divided by one:
You have zero cookies to share equally among one child, how many cookies does that child get?
Now comes the philosophical part:
one divided by zero:
You have one cookie to share equally among zero children, how many cookies does each child get?
Claude
It is good to 'make sense' out of the choices so that you don't have to rely on memory. It is even better if the kids can make sense out of it! (Surrendered K... ๐)
CB 
Kaustubh KatdareOne cookie given to zero children. Each gets zero cookies , woa! So one divided by zero is zero

Raviteja.gyes as crazyboy said that infinity is a countless number and undefined is a number we know but we can not define which is correct
right?
but as per above explanation number divided with zero is zero
what a logical explanation! 
pooja sehgaldividing by zero we get infinity in mathematics
actually it is x/0=approaches to infinity
and in maths infinity is ( largest no of items that can b defined)
it is different from other real numbers, not exactly defined
so in terms of maths it is not defined
but in other philosiphical, logical terms one can reach to different meaning as explained above 
Anil JainThe_Big_KOne cookie given to zero children. Each gets zero cookies , woa! So one divided by zero is zero
Wow Biggi... ๐
you Got it ๐ that, I was talking about invisible man who do not eat cookies... ๐, thats why cookies= 0

vipandeepLets try it this way.
x = a/b or xb=a we need to find x right?
now if b = 0 x.0=a 0=a
now two cases arise
1. if a = 0
if a = 0 then we can have any number as a solution of xb=a because no matter what value is x it doesn't matter since x.0 = a where a = 0
2. if a not = 0
then we have problems. x.0=a where a is not 0. we need to find a number x for which the x.0 becomes a. but there is not way we can get x so that x.0 equals a nonzero number.
such number does not exist.
so it is undefined. 
Ankita Katdare@Vipandeep: That's a good explanation. So, it is clear that undefined and infinity are two different things, right?

silverscorpionHmm.. Let me try something..
Let's say we have a number 5. Divide it by 10. You get 0.5.
If you divide it by 0.1, you get 50. Going on, we have,
5/0.1 = 50; 5/.01 = 500; 5/.001 = 5000 etc..
We see that as the denominator approaches zero, the value of the division becomes larger and larger.
So, when the denominator actually reaches zero, then the division will give you a very very large number.
We call that very large number by the name infinity. The exact value of infinity is not defined. But infinity itself is very well defined.
Now, undefined is something else.
Take the expression 0/0. What will it give? You have,
1) 0 divided by any number is 0
2) any number divided by 0 is infinity
3) any number divided by the same number is 1
Now, 0/0 is 0 or infinity or 1?? This is undefined.
So, expressions like 0/0. 0/infinity, infinity/0, 0^0, 0^infinity etc.. are undefined.
But infinity is defined. ๐๐ 
Ankita Katdare
Very clearly stated. Thank you. It is very clear now. ๐silverscorpionWe see that as the denominator approaches zero, the value of the division becomes larger and larger.
So, when the denominator actually reaches zero, then the division will give you a very very large number.
We call that very large number by the name infinity. The exact value of infinity is not defined. But infinity itself is very well defined.
Now, 0/0 is 0 or infinity or 1?? This is undefined.
So, expressions like 0/0. 0/infinity, infinity/0, 0^0, 0^infinity etc.. are undefined.
But infinity is defined. ๐๐ 
ISHAN TOPREYeah.You have explained the meaning of life. lol :razz:

silverscorpion
Hmm.. All hail me!!ishutopreYeah.You have explained the meaning of life. lol :razz:
By the way, on an unrelated note, the answer to life, the universe and everything is 42. ๐ 
Deepika BansalSS: That's a well stated answer. Thank you for clearing the difference between infinity and undefined.

ajayk2Yes 'infinity' and 'undefined' are two different notions. First of all, infinity is not a number, it's an idea but the idea is well established. Since infinity is not a not a number, the usual algebra can't be applied on it. Also I'd like to point here the notion of undefined in case of division by zero.
(i) 0/0 is undefined. Let 0/0 = x. => 0 X x = 0. Any value of x (real or complex, will satisfy this equation). So x is undefined because of lack of uniqueness. Recall that the algebraic operations of addition and multiplication are binary functions. The result value is always unique.
(ii) a/0 is undefined (a is a nonzero number, real or complex) Let a/0 = y. => a = 0 X y. No value of y can satisfy this equation because any number multiplied by zero is always zero. (Is it an assumption?) So it's undefined because of a lack of solution.
Now I've a question here.
We say tan 90 is infinity. Why? Shouldn't it undefined because cos 90 = 0. 
ajayk2Yes 'infinity' and 'undefined' are two different notions. First of all, infinity is not a number, it's an idea but the idea is well established. Since infinity is not a not a number, the usual algebra can't be applied on it. Also I'd like to point here the notion of undefined in case of division by zero.
(i) 0/0 is undefined. Let 0/0 = x. => 0 X x = 0. Any value of x (real or complex, will satisfy this equation). So x is undefined because of lack of uniqueness. Recall that the algebraic operations of addition and multiplication are binary functions. The result value is always unique.
(ii) a/0 is undefined (a is a nonzero number, real or complex) Let a/0 = y. => a = 0 X y. No value of y can satisfy this equation because any number multiplied by zero is always zero. (Is it an assumption?) So it's undefined because of a lack of solution.
Now I've a question here.
We say tan 90 is infinity. Why? Shouldn't it undefined because cos 90 = 0. 
silverscorpionajayk2Yes 'infinity' and 'undefined' are two different notions. First of all, infinity is not a number, it's an idea but the idea is well established. Since infinity is not a not a number, the usual algebra can't be applied on it. Also I'd like to point here the notion of undefined in case of division by zero.
(i) 0/0 is undefined. Let 0/0 = x. => 0 X x = 0. Any value of x (real or complex, will satisfy this equation). So x is undefined because of lack of uniqueness. Recall that the algebraic operations of addition and multiplication are binary functions. The result value is always unique.
(ii) a/0 is undefined (a is a nonzero number, real or complex) Let a/0 = y. => a = 0 X y. No value of y can satisfy this equation because any number multiplied by zero is always zero. (Is it an assumption?) So it's undefined because of a lack of solution.
Now I've a question here.
We say tan 90 is infinity. Why? Shouldn't it undefined because cos 90 = 0.
Hi Ajay,
Refer to my post above.
a/0 is not undefined. It's what we call infinity. 0/0 is undefined.
And tan 90 is infinity precisely because cos 90 is zero.. 
ajayk2Yes I read your explanation but my point is that way the denominator would be very close to zero but it will never be equal to zero. You are talking about the concept of limit. It's okay to write
lim (tan x) = inf. but tan 90 = inf is wrong I think. so lim a/x has got meaning but
x> 90 x>0
a/0 is undefined and hence meaningless. Tell me what you think. 
Nobody_If n is any number.
n/0 = undefined
(going by the cookie story..if u have n cookies and u give them to no one, then you simply cannot determine how many each will receive..it's sort of meaningless, hence undefined)
0/0
this has 2 theories:
a) undefined (following the same cookie story; the division being pointless, and thus undefined)
b) Infinity! (because 0 multiplied by anything is 0, following rules of multiplication where both the dividend and divisor are 0)
0/n = 0
(any number multiplied with 0 gives 0) 
Ramani AswathI missed this post. Since school days I have been fascinated by this. Actually there are many infinities some of which are demonstrably larger than others.
This article sheds some (darkish!) light on this:
Strange but True: Infinity Comes in Different Sizes  Scientific American
In his entertaining book:'One, Two, Three Infinity, George Gamov has a chapter on this. I strongly recommend this book to all. According to him there are three infinities, each of a different size. 
Shashank Moghe
Wrong question. There is no child to begin with. It is like being asked to write one of those essays "If I were not born". If I were not born, I would not be sitting in this school listening to a teacher asking me to write an essay.Anil Jainone divided by zero:
You have one cookie to share equally among zero children, how many cookies does each child get? 
Shashank Moghe
If you try that on a scientific calculator, it should give you NaN (Not a Number). It is not "infinity". Tan(90) is just "not defined".ajayk2Yes 'infinity' and 'undefined' are two different notions. First of all, infinity is not a number, it's an idea but the idea is well established. Since infinity is not a not a number, the usual algebra can't be applied on it. Also I'd like to point here the notion of undefined in case of division by zero.
(i) 0/0 is undefined. Let 0/0 = x. => 0 X x = 0. Any value of x (real or complex, will satisfy this equation). So x is undefined because of lack of uniqueness. Recall that the algebraic operations of addition and multiplication are binary functions. The result value is always unique.
(ii) a/0 is undefined (a is a nonzero number, real or complex) Let a/0 = y. => a = 0 X y. No value of y can satisfy this equation because any number multiplied by zero is always zero. (Is it an assumption?) So it's undefined because of a lack of solution.
Now I've a question here.
We say tan 90 is infinity. Why? Shouldn't it undefined because cos 90 = 0. 
Shashank Moghe
Division by zero is not defined. Any hypotheses based on the presumption of the kind x/b = a where b = 0, are making a fundamental mistake of assuming there exists the RHS 'a'. It does not, because the operation is not defined!silverscorpionHi Ajay,
Refer to my post above.
a/0 is not undefined. It's what we call infinity. 0/0 is undefined.
And tan 90 is infinity precisely because cos 90 is zero..
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